Organizing centers in one-dimensional, piecewise-linear maps
BV-Forschungsberichte 2014,2
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Beschreibung:
The possible dynamics of one dimensional, piecewise-linear, discontinuous systems can be varied and very complex. Just a small variation in the parameters can lead to completely different bifurcations. This creates the problem of effectively investigating such a system. The solution that has been found in previous works is to investigate so-called organizing centers.
An organizing center is a point with high codimension in the parameter space of a dynamical system. It has the property that a large number of bifurcation curves intersect there. Usually these bifurcation curves do not exist only near the organizing center, but cover a large part of the parameter space. This means that a large part of the parameter space contains the same bifurcations than the neighborhood of the organizing center. So, to understand this large part of the parameter space it is sufficient to investigate the neighborhood of the organizing center. The consequence of that is that the parameter space of a dynamical system can be investigated very effectively by first finding organizing centers in it and by then investigating the neighborhoods of these organizing centers. The result of such an investigation is one or several bifurcation scenarios, that the organizing center is said to "produce".
In the past this was done for several specific one dimensional, piecewise-linear, discontinuous systems with one point of discontinuity (two-partition systems). For each of these specific systems the organizing centers were identified and the bifurcation structures that they produce were investigated. In contrast to these previous works that investigated specific two-partition systems this work now presents a comprehensive overview over the organizing centers of piecewise-linear two-partition systems in general, thus presenting an overview over all asymptotic dynamics that are possible in these systems. Note that there is strong evidence that the results of this work also apply to systems with one point of discontinuity which are not piecewise-linear but piecewise-smooth.
Two partition systems have in the general case four parameters: the offsets and the slopes. It is shown in this work that all organizing centers of these systems can be described by setting the slopes to fixed values and then describing the resulting offset plane. Furthermore, the possible slope combinations can be grouped into 16 different cases, so that the organizing centers that correspond to a specific case all produce similar bifurcation scenarios. These 16 cases can be reduced to 10 cases due to topological conjugacy.
In addition to the description of organizing centers in piecewise-linear two-partition systems, this work also contains an investigation of organizing centers in one dimensional, piecewise-linear, discontinuous systems with two points of discontinuity (three-partition systems). This investigation is less comprehensive than the description of the dynamics of two-partition systems. The reason for that is that a full description of the dynamics of three-partition systems would be beyond the scope of this work. Instead, the investigation concentrates on a specific type of organizing center which produces a specific bifurcation scenario, the socalled nested period incrementing (NPI) scenario.
The NPI scenario appears in piecewise-linear three-partition systems that contain a positive slope, a negative slope and a slope that is zero. Depending on the specific system it can appear in a fully and a partially developed form. The fully developed NPI scenario contains a specific set of periodic orbits. The partially developed NPI scenario only contains a subset of these periodic orbits. Both forms of the scenario are described in detail.
An organizing center is a point with high codimension in the parameter space of a dynamical system. It has the property that a large number of bifurcation curves intersect there. Usually these bifurcation curves do not exist only near the organizing center, but cover a large part of the parameter space. This means that a large part of the parameter space contains the same bifurcations than the neighborhood of the organizing center. So, to understand this large part of the parameter space it is sufficient to investigate the neighborhood of the organizing center. The consequence of that is that the parameter space of a dynamical system can be investigated very effectively by first finding organizing centers in it and by then investigating the neighborhoods of these organizing centers. The result of such an investigation is one or several bifurcation scenarios, that the organizing center is said to "produce".
In the past this was done for several specific one dimensional, piecewise-linear, discontinuous systems with one point of discontinuity (two-partition systems). For each of these specific systems the organizing centers were identified and the bifurcation structures that they produce were investigated. In contrast to these previous works that investigated specific two-partition systems this work now presents a comprehensive overview over the organizing centers of piecewise-linear two-partition systems in general, thus presenting an overview over all asymptotic dynamics that are possible in these systems. Note that there is strong evidence that the results of this work also apply to systems with one point of discontinuity which are not piecewise-linear but piecewise-smooth.
Two partition systems have in the general case four parameters: the offsets and the slopes. It is shown in this work that all organizing centers of these systems can be described by setting the slopes to fixed values and then describing the resulting offset plane. Furthermore, the possible slope combinations can be grouped into 16 different cases, so that the organizing centers that correspond to a specific case all produce similar bifurcation scenarios. These 16 cases can be reduced to 10 cases due to topological conjugacy.
In addition to the description of organizing centers in piecewise-linear two-partition systems, this work also contains an investigation of organizing centers in one dimensional, piecewise-linear, discontinuous systems with two points of discontinuity (three-partition systems). This investigation is less comprehensive than the description of the dynamics of two-partition systems. The reason for that is that a full description of the dynamics of three-partition systems would be beyond the scope of this work. Instead, the investigation concentrates on a specific type of organizing center which produces a specific bifurcation scenario, the socalled nested period incrementing (NPI) scenario.
The NPI scenario appears in piecewise-linear three-partition systems that contain a positive slope, a negative slope and a slope that is zero. Depending on the specific system it can appear in a fully and a partially developed form. The fully developed NPI scenario contains a specific set of periodic orbits. The partially developed NPI scenario only contains a subset of these periodic orbits. Both forms of the scenario are described in detail.